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A New Symmetry of the Colored Alexander Polynomial ITEP/TH-04/20 IITP/TH-04/20 MIPT/TH-04/20 A new symmetry of the colored Alexander polynomial V. Mishnyakov a,c,d,∗ A. Sleptsova,b,c,† N. Tselousova,c‡ a Institute for Theoretical and Experimental Physics, Moscow 117218, Russia b Institute for Information Transmission Problems, Moscow 127994, Russia c Moscow Institute of Physics and Technology, Dolgoprudny 141701, Russia d Institute for Theoretical and Mathematical Physics, Moscow State University, Moscow 119991, Russia Abstract We present a new conjectural symmetry of the colored Alexander polynomial, that is the specializa- tion of the quantum slN invariant widely known as the colored HOMFLY-PT polynomial. We provide arguments in support of the existence of the symmetry by studying the loop expansion and the character expansion of the colored HOMFLY-PT polynomial. We study the constraints this symmetry imposes on the group theoretic structure of the loop expansion and provide solutions to those constraints. The symmetry is a powerful tool for research on polynomial knot invariants and in the end we suggest several possible applications of the symmetry. 1 Introduction The colored HOMFLY polynomial is a topological link invariant. It attracts a lot of attention because it is connected to various topics in theoretical and mathematical physics: quantum field theories [5, 6, 7], quantum groups [11, 12, 8], conformal field theories [9], topological strings [10]. Whenever an explicit calculation of a class of HOMFLY invariants is derived it causes advancements in these areas. However, the computations be- come extremely difficult as the representation gets larger. At present moment explicit expressions are available only for several classes of knots and representations, including symmetric and anti-symmetric representations. K In this paper we mainly discuss the symmetry of the colored Alexander polynomial AR(q) (20), which is a special case of colored HOMFLY polynomial. We conjecture a new "tug-the-hook" symmetry (25), that relates two Alexander polynomials with different representations. One of peculiar properties of this symme- try is that it connects rectangular representations with non-rectangular ones. We also conjecture that this arXiv:2001.10596v4 [hep-th] 16 Jul 2020 symmetry is valid for any knot. Let us proceed to the symmetry itself. The first nontrivial example of the action of the symmetry, that cannot be reduceв to the previously known symmetries, involves sufficiently large representations. As far as no explicit results are known for large representations except Rosso-Jones formula for torus knots [20] we checked the symmetry only in that case. For example: 31 31 A[4,3](q)= A[3,2,2](q) (1) 31 31 A[4,4](q)= A[3,3,2](q) (2) 31 31 A[5,4,4](q)= A[4,3,3](q) (3) ∗[email protected] †[email protected] ‡[email protected] 1 To make our statements more reasonable let us provide some arguments in support of this symmetry. These arguments are somehow knot-independent and additionally justify verifying the symmetry in the case of torus knots. • It is well known fact that Alexander polynomial arises in 3d Chern-Simons theory with superalgebra sl(1|1) [36, 37, 38, 39]. Carefull consideration of the representation theory of superalgebras reveals that Young diagrams enumerate the representations ambiguously. In general for a representation one can align several Young diagrams, that are connected via some relation. If this relation is applied to the representation of the K colored Alexander polynomial AR(q) one obtains the tug-the-hook symmetry. For more careful consideration of this argument see [33]. • The tug-the-hook symmetry together with the rank-level duality explains vanishing of 1,3,5 orders in the loop expansion of the colored Alexander polynomial for arbitrary knot (see Section 5.4). This fact is surely known from the trivalent diagram point of view. All trivalent diagrams at these levels are known and we can establish the vanishing of the corresponding group-factors. However it is not obvious a priori, without calculating the trivalent diagrams explicitly. Also we see that 2,4,6 orders of the loop expansion have the structure compatible with the symmetry. • The symmetry is tightly connected with the eigenvalue conjecture [26]. One of the possible formulations of the conjecture is the set of quantum Ri-matrices is completely determined by the normalized eigenvalues of the quantum universal Rˇ-matrix. The eigenvalue conjecture has been checked in nu- merous cases and proven in some of them (see Section 2.3 in [30] for a review of checks of the eigenvalue conjecture). In Section 6 we have shown that the symmetry is a corollary of the eigenvalue conjecture. It actually also strengthens our examples (1,2,3). In Section 6 we observe that symmetry follows from con- servation of the R matrices eigenvalues, hence the knot itself does not appear in the proof. Hence verifying it for some knots leads to conclusion that it is true for all knots. We would like to emphasize that the origin of this symmetry can be described from three different points of view: Lie superalgebras, quantum Lie algebras, and classical Lie algebras. In each case, it follows from the invariance of a certain algebraic structure. The paper is organized as follows. In Section 2 we review some basic facts about the colored HOMFLY. In Section 3 we define the colored Alexander polynomial and discuss the motivation to study it. In Section 4 we present a new conjectural symmetry of the colored Alexander polynomial. We define the tug-the-hook solutions in Section 5 to study the loop expansion. Full description of the tug-the-hook solutions is presented in Section 5.1 in terms of combinatorics. In Section 5.2 we discuss derivation of the solutions. Section 5.4 is devoted to the explanation of the form of the loop expansion of the colored Alexander with the help of the symmetry. In Section 6 we discuss the connection of the symmetry with the eigenvalue conjecture. The last Section 7 is devoted to possible applications of the tug-the-hook symmetry. 2 The colored HOMFLY polynomial The HOMFLY polynomial can be arrived in various ways. We mention two of them: • Chern-Simons theory approach. The colored HOMFLY polynomial of a knot K can be obtained as the average of the Wilson loop in Chern-Simons theory with the gauge group SU(N) on S3 [5, 15]. In our notation R stands for a representation of the gauge group and in particular case of SU(N) is enumerated by a Young diagram. K HR (q,a)= trR P exp A , (4) IK CS where Chern-Simons action is given by κ 2 SCS[A]= tr A ∧ dA + A ∧ A ∧ A . (5) 4π 3 3 ZS The polynomial variables are parameterized as follows: ~ ~ 2πi q = e , a = eN , ~ := . (6) κ + N One can evaluate (4) in the holomorphic gauge Ax + iAy = 0 [18] and obtain the loop expansion of the 2 HOMFLY polynomial [15] ∞ K K R ~n HR (q,a)= vn,mrn,m . (7) n=0 m ! X X A remarkable fact about this expansion is that the knot and group dependence splits. Knot dependent K parts vn,m are integrals of fields’ averages along the loop. They are known as Vassiliev invariants of knots or R sl invariants of finite-type. Group dependent parts rn,m are called group factors and are traces of N generators Ti. R rn,m ∼ trR T (m) T (m) ...T (m) (8) i1 i2 i2n (m) (m) (m) K i1 i2 in vn,m ∼ dx1 dx2 ... dxn A (x1) A (x2) ...A (x2n) (9) I Z Z D E If one proceeds with several extra transformations of (7) in the holomorphic gauge, one can arrive the Kontsevich integral form [18, 21]. Before turning to this particular form let us review some basic facts about the Kontsevich integral and Lie algebra weight systems [22] (see Chapter 6). Full definition of the Kontsevich integral can be found in [22] (see Chapter 8.2). However, what matters for us is that its values belong to the graded completion of the algebra of unframed chord diagrams Dˆ [22] (see Chapter 4) ∞ Z(K)= V(K)n,m Dn,m. (10) n=0 m X X We denote Dn,m a chord diagram with n chords and V(K)n,m the coefficient of the chord diagram in the Kontsevich integral. Some examples of the unframed chord diagrams in small degrees are: D0,1 D2,1 D3,1 D4,1 D4,2 D4,3 The Lie algebra weight system ϕslN is the homomorphism from the algebra of unframed chord diagrams D to the center of the universal enveloping algebra ZU(slN ). The definition of ϕ and the proof of this statement can be found in [22] (see Chapter 6). The mapping ϕslN is clear from examples. a b* a c* b b* b a* a* c dim slN dim slN ∗ ∗ ∗ ∗ ∗ ϕslN (D2,1)= TaTbTa Tb ϕslN (D3,1)= TaTbTa TcTb Tc a,b a,b,c X=1 X=1 Choosing an irreducible representation of slN identified with the Young diagram R we look at the Lie algebra weight system associated with the representation R ϕsl R N sl ρR Tr C ϕslN : D −−−→ ZU( N ) −−→ End(V ) −→ (11) R Linearly extending the action of ϕslN we obtain that the loop expansion of the colored HOMFLY polynomial is the special value of the Kontsevich integral R K ϕslN (Z(K)) = HR (q,a) (12) • Reshetikhin-Turaev approach. Knots and links are tightly connected to braids through the Alexan- der’s theorem. It states that every knot or link can be represented as a closure of a braid.
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